The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

Abstract

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma -which provides a sufficient condition for this to occur -corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37,38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

Sufficient Conditions for Uniform Bounds in Abstract Polymer Systems and Explorative Partition Schemes

“…There is a deep connection between the Lovász Local Lemma [5] and the partition function of the abstract polymer system at negative real fugacity discovered by Scott & Sokal [13]. As already done for Fernández & Procacci's SCUB [2], the new SCUBs also improve the Lovász Local Lemma.…”

“…The proof is inductiveà la Dobrushin [4] and an extension of the homogeneous version stated in [13,Corollary 5.7]. The induction over the cardinality of finite subsets Λ of P and an ordering of the polymers ensures a bound of ϕ γ Λ without ever considering I (γ) \ Λ = ∅.…”

“…Here, we do not want to reuse the last visited different label, which is a similar effect as the escaping pairs in the section 4.4. More general, when one looks at the inductive proof of the reduced SCUB in section 3, then one wants to transfer the following property: labels are at most used once along each path in the inductive derivation tree (see also [13,Algorithm T]). If we translate this into the cluster expansion setting, then we want the polymer labels of a path from the root in a singleton tree to form a lazy self-avoiding walk in the polymer system.…”

“…There is another Dobrushin-style SCUB building on work of Scott & Sokal [13], which reduces the degree of Dobrushin's SCUB by one. It is better than Fernández & Procacci's SCUB on triangle-free graphs and optimal on trees, but may be worse on general graphs.…”

The Lee‐Yang and Pólya‐Schur programs. II. Theory of stable polynomials and applications

“…By slight abuse of terminology, if Ψ ⊂ C and Ω = Ψ n then Ω-stable polynomials will also be referred to as Ψ-stable. In physics [54,56] it is useful to distinguish between hard-core pair interactions (subject to constraints, e.g. the maximum degree of a graph) and soft-core pair interactions (essentially constraint-free).…”